Ecole Normale Supérieure de Lyon,
The theory of stochastic processes offers both a formalism and a technique that spans many disciplines, addressing questions ranging from the fundamental (e.g., the origin of irreversibility) to the very applied (e.g., the elimination of noise in signals). In statistical mechanics, fluid mechanics, and related fields in physics and chemistry, this theory complements microscopic and macroscopic descriptions by lumping, on the one hand, a large number of degrees of freedom into a stochastic force or, on the other hand, by refining the macroscopic laws with corrections due to finite size or discreteness of the underlying microscopic processes. Brownian motion, light scattering, fluctuating hydrodynamics and, more generally, the theory of Gaussian stochastic processes and its application to equilibrium fluctuations are classical examples of the power and success of this approach.Besides its well established position in statistical mechanics close to equilibrium, the field of stochastic processes has, to a certain extent parallel with the developments in dynamical systems and chaos theory, received a novel impetus over the last 20 years. Several new noise-induced phenomena have been discovered and others are being reassessed. We cite: stochastic resonance, involving the amplification of a time periodic signal by the addition of a resonant amount of noise; Brownian motors, which operate by rectifying non-thermal noise using ratchet-like potentials; noise induced phase transitions and more generally noise induced effects in spatially distributed systems, in which nonequilibrium noise generates symmetry breaking phase transitions including the appearance of spatial or temporal patterns; and, on the more fundamental side, the advances in the microscopic understanding of irreversibility by the mapping of reversible microscopic dynamics onto an irreversible stochastic description based on chaos-like sensitivity to initial conditions.
Numerical methods, including computer simulations, have played an important role in these developments. The combination of noise with nonlinearity, the absence of detailed balance or the large number of degrees of freedom implied in these problems makes a full and detailed analytic treatment of all the issues difficult or impossible. Surprisingly, no organized scientific activity (like sponsored meetings, collaboration programs, or contact groups) is in place to channel or coordinate the different efforts, especially those related to computational issues.
There have been several meetings on topics in stochastic processes (for example noise in nuclear reactors, 1/f noise, stochastic resonance or Brownian motors), but never one in which computational issues formed the connecting theme. Our hope is that the proposed meeting, joining experts in either computational or theoretical aspects of stochastic processes belonging to widely different research topics, will provide a forum in which both numerical and theoretical issues in stochastic processes are discussed in a context that supersedes the details of the underlying research topics. By confrontation and exchange about the various specific numerical methods, we hope to make progress on several technical issues (time-marching methods for stochastic ordinary and partial differential equations, the effect of noise on numerical stability, direct integration of Fokker Planck equations, improved simulation algorithms for discrete Markov chains, parallel methods for large scale simulations, etc.). By inviting experimental groups, working on the effect of noise in reaction diffusion systems, we hope to increase the direct confrontation of experiment with numerical simulations of approximate models. The meeting should also stimulate further theoretical interest and progress in topics of current interest, in particular noise induced instabilities, their possible link to energy concentration in nonlinear systems, Brownian motors, and the nature of irreversibility. Finally and foremost, we hope to forster or renew collaborations between participants including a better coordination and exchange of the respective computational capabilities. We plan to organize a related tutorial meeting to facilitate further the transfer of expertise.